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Dec 3, 2025
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Understanding the Standard Normal Distribution - Lesson 4
calista ๐ชป
@urstrulycalista
1 / 3
Standard Normal Distribution and Z-Scores
Ever wonder how to compare values from different datasets? Z-scores are your answer! A standard normal distribution transforms regular data into standardized values called z-scores.
Z-scores tell you exactly how many standard deviations a value is from the mean. If a value is larger than the mean, it has a positive z-score; if smaller, it has a negative z-score. When a value equals the mean, its z-score is exactly zero.
To calculate a z-score, use the formula:
Z = (X - ฮผ)/ฯ
where X is your value, ฮผ is the mean, and ฯ is the standard deviation.
Pro Tip: Z-scores make comparing values from completely different datasets possible because they convert everything to the same scale of standard deviations from the mean.
Calculating Z-Scores: Examples
Let's put z-scores to work! Imagine we have a normal distribution with mean ฮผ = 5 and standard deviation ฯ = 6. If we observe X = 17, what's the z-score?
Z = (X - ฮผ)/ฯ = (17 - 5)/6 = 2
This means 17 is positioned exactly 2 standard deviations above the mean of 5. That's pretty far out in the distribution!
Now let's try X = 1:
Z = (X - ฮผ)/ฯ = (1 - 5)/6 = -0.67
The negative z-score tells us that 1 is 0.67 standard deviations below the mean. You'll quickly notice that z-scores give you an immediate sense of how unusual a value is in your dataset.
The Empirical Rule
The Empirical Rule is your shortcut for understanding normal distributions! It tells us how data is distributed around the mean in a bell-shaped curve.
For normally distributed data, approximately:
- 68% of values fall within 1 standard deviation of the mean (ฮผ ยฑ 1ฯ)
- 95% of values fall within 2 standard deviations (ฮผ ยฑ 2ฯ)
- 99.7% of values fall within 3 standard deviations (ฮผ ยฑ 3ฯ)
This rule is incredibly useful when making quick estimations about your data. For example, if test scores are normally distributed with mean 75 and standard deviation 5, you can quickly determine that about 95% of students scored between 65-85.
Remember: The Empirical Rule only works for bell-shaped, symmetric distributions. Always check your data's shape before applying it!
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What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
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You can download the app in the Google Play Store and in the Apple App Store.
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think itโs very much worth it and youโll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still donโt get it!!! In the end I think itโs worth it ๐๐ โ ๏ธAlso DID I MENTION ITS FREEE YOU DONโT HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLYโ๏ธโ๏ธโ ๏ธ
Thomas R
iOS user
Knowunity is the BEST app Iโve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help๐ค๐ค
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.๐๐ฉท๐
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH ๐๐๐ฒ๐ค๐โจ๐๐ฎ
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as Iโve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think itโs very much worth it and youโll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still donโt get it!!! In the end I think itโs worth it ๐๐ โ ๏ธAlso DID I MENTION ITS FREEE YOU DONโT HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLYโ๏ธโ๏ธโ ๏ธ
Thomas R
iOS user
Knowunity is the BEST app Iโve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help๐ค๐ค
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.๐๐ฉท๐
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH ๐๐๐ฒ๐ค๐โจ๐๐ฎ
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as Iโve seen! Geometry too!
Paul T
iOS user
Understanding the Standard Normal Distribution - Lesson 4
calista ๐ชป
@urstrulycalista
In statistics, understanding how data is distributed and measured is crucial. This week covers normal distributions, z-scores, and the Central Limit Theoremโessential concepts that help us analyze data patterns and make predictions based on probability distributions.
Standard Normal Distribution and Z-Scores
Ever wonder how to compare values from different datasets? Z-scores are your answer! A standard normal distribution transforms regular data into standardized values called z-scores.
Z-scores tell you exactly how many standard deviations a value is from the mean. If a value is larger than the mean, it has a positive z-score; if smaller, it has a negative z-score. When a value equals the mean, its z-score is exactly zero.
To calculate a z-score, use the formula:
Z = (X - ฮผ)/ฯ
where X is your value, ฮผ is the mean, and ฯ is the standard deviation.
Pro Tip: Z-scores make comparing values from completely different datasets possible because they convert everything to the same scale of standard deviations from the mean.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Calculating Z-Scores: Examples
Let's put z-scores to work! Imagine we have a normal distribution with mean ฮผ = 5 and standard deviation ฯ = 6. If we observe X = 17, what's the z-score?
Z = (X - ฮผ)/ฯ = (17 - 5)/6 = 2
This means 17 is positioned exactly 2 standard deviations above the mean of 5. That's pretty far out in the distribution!
Now let's try X = 1:
Z = (X - ฮผ)/ฯ = (1 - 5)/6 = -0.67
The negative z-score tells us that 1 is 0.67 standard deviations below the mean. You'll quickly notice that z-scores give you an immediate sense of how unusual a value is in your dataset.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The Empirical Rule
The Empirical Rule is your shortcut for understanding normal distributions! It tells us how data is distributed around the mean in a bell-shaped curve.
For normally distributed data, approximately:
- 68% of values fall within 1 standard deviation of the mean (ฮผ ยฑ 1ฯ)
- 95% of values fall within 2 standard deviations (ฮผ ยฑ 2ฯ)
- 99.7% of values fall within 3 standard deviations (ฮผ ยฑ 3ฯ)
This rule is incredibly useful when making quick estimations about your data. For example, if test scores are normally distributed with mean 75 and standard deviation 5, you can quickly determine that about 95% of students scored between 65-85.
Remember: The Empirical Rule only works for bell-shaped, symmetric distributions. Always check your data's shape before applying it!
We thought youโd never ask...
What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help โ all at your fingertips.
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4.9/5
App Store
4.8/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think itโs very much worth it and youโll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still donโt get it!!! In the end I think itโs worth it ๐๐ โ ๏ธAlso DID I MENTION ITS FREEE YOU DONโT HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLYโ๏ธโ๏ธโ ๏ธ
Thomas R
iOS user
Knowunity is the BEST app Iโve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help๐ค๐ค
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.๐๐ฉท๐
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH ๐๐๐ฒ๐ค๐โจ๐๐ฎ
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as Iโve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think itโs very much worth it and youโll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still donโt get it!!! In the end I think itโs worth it ๐๐ โ ๏ธAlso DID I MENTION ITS FREEE YOU DONโT HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLYโ๏ธโ๏ธโ ๏ธ
Thomas R
iOS user
Knowunity is the BEST app Iโve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help๐ค๐ค
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.๐๐ฉท๐
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH ๐๐๐ฒ๐ค๐โจ๐๐ฎ
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as Iโve seen! Geometry too!
Paul T
iOS user